In anime, fanservice stands for scenes which are not relevant to the overall plot at all, but merely are included to give the viewers what they presumably want to see. This generally includes things like panty shots, bouncing boobs, shower scenes, skirts which barely cover the crotch area, but can also be things like incredibly overexaggerated explosions, giant robots or supercute animals.

Do an image search for "Agent Aika" and you will see. That series basically is ONLY fanservice without a plot.

In general, fan service refers to scenes designed to excite or titillate the viewer. This can include scantily-clad outfits, cleavage shots, panty shots, nude scenes (shower scenes especially), etc. Some broader definitions also include things like cool mecha, big explosions, battle scenes, etc. Basically, if it has little plot-redeaming value, but makes the viewer sit up and take notice, it's probably fan service in one form or another.

Usually seen in reference to Japanese popculture. When someone (i.e. manga/anime character, music artist) flashes the general public and/or whenever two or more famous people engage in any action which could be perceived as sexual in nature just to stand out or please (excite, shock, etc.) their fans. Many bandmembers have been known to kiss, lick, rub, and hump against other bandmembers while performing or while in public. Usually, this is purely platonic, but not always.

Person #1: Did you see the ~ *insert band name here* performed at their last concert?
Person #2: Where the vocalist licked the bassist's neck, and later on the bassist humped the vocalist?
Person #1: Yeah, wasn't it great?!

As stated above, "fanservice" came from Japanese anime, film, and music culture. Its definition is spreading to include any unnecessary visuals included simply because the fans, presumably, want to see them. Gratuitous nudity, male or female, but also shots of cute mascot animals (e.g., Chuchu the mouse in "Utena").

In terms of Anime, the least extreme form of nudity or sexually based angles. This may include panty shots, minor nudity, underwear shots, bouncing breasts and other such things added solely for comic effect or pleasuring perverted fans.

Generally any situation or scene in a movie, cartoon, comic, or even real life in which a scene is done specifically to cater to what the fans want to see, divorced from a plot or any other non-catering need. Often mistaken as invented by Japanese anime, the truth of the matter is that Hollywood movies and American TV shows in particular had been using this concept for years, if not decades, before anime ever even existed; however, the coined word itself DID start within anime fandoms to reference scenes of near-nudity that held no part to the overall plot of the episode/movie/etc.

The first film to involve nudity in the US was 1959, whereas the advent of the modern style and tradition of anime was in the late 60's, hence fanservice existed prior to anime.